withr::with_dir(here::here(), {
  targets::tar_load(c(
    model_data_1ha, model_data_cf
    ))
})

Purpose

In GAMs, the number of knots used for a smooth needs to be great enough to capture the “true” complexity of a relationship. However, higher number of knots increase computational demand. Here I’ll do some diagnostics and trial-and-error to select an adequate number of knots. I’m going to use models without random effects for computational efficiency.

Survival

Continuous forest

s_cf <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(5, 18),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_cf,
  method = "fREML",
  select = TRUE
)
set.seed(888)
k.check(s_cf)
##                   k'       edf   k-index p-value
## s(log_size_prev)   9  3.467671 0.9622796  0.9675
## s(spei_history,L) 72 13.648312        NA      NA

Looks like the default k is adequate for s(log_size_prev). Unfortunately k.check() doesn’t work for data with matrices, so I’ll use an alternative approach detailed in ?mgcv::choose.k. Briefly, it involves looking for pattern in residuals.

source(here::here("R", "utils.R"))
check_res_edf(s_cf)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)   1.8

We want this number to be close to 0. First let’s up the lag dimension number of knots.

s_cf2 <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(5, 30),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_cf,
  method = "fREML",
  select = TRUE
)
edf(s_cf)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.47
## 2 s(spei_history,L) 13.6
edf(s_cf2)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.47
## 2 s(spei_history,L) 13.7
draw(s_cf, select = "s(spei_history,L)")
## Warning: Removed 939 rows containing non-finite values (stat_contour).

draw(s_cf2, select = "s(spei_history,L)")
## Warning: Removed 939 rows containing non-finite values (stat_contour).

Didn’t change edf at all really, or shape of smooth, so back to original k for lag and up k for spei

s_cf3 <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(15, 18),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_cf,
  method = "fREML",
  select = TRUE
)
edf(s_cf)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.47
## 2 s(spei_history,L) 13.6
edf(s_cf3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.47
## 2 s(spei_history,L) 15.5
draw(s_cf, select = "s(spei_history,L)")
## Warning: Removed 939 rows containing non-finite values (stat_contour).

draw(s_cf3, select = "s(spei_history,L)")
## Warning: Removed 939 rows containing non-finite values (stat_contour).

EDF went up and shape of smooth changed, so check if its enough.

check_res_edf(s_cf3)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)   1.1

Can I reduce number of knots in the lag dimension?

s_cf4 <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(15, 15),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_cf,
  method = "fREML",
  select = TRUE
)
edf(s_cf3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.47
## 2 s(spei_history,L) 15.5
edf(s_cf4)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.47
## 2 s(spei_history,L) 15.5
draw(s_cf3, select = "s(spei_history,L)")
## Warning: Removed 939 rows containing non-finite values (stat_contour).

draw(s_cf4, select = "s(spei_history,L)")
## Warning: Removed 939 rows containing non-finite values (stat_contour).

Shape is unchanged

check_res_edf(s_cf4)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)   1.1

Looks good. Final knots for s_cf are:

log_size_prev: 10 spei_history, L: 15,15

rm(s_cf, s_cf2, s_cf3, s_cf4)

Fragment

s_1ha <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(5, 18),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_1ha,
  method = "fREML",
  select = TRUE
)
set.seed(888)
k.check(s_1ha)
##                   k'      edf   k-index p-value
## s(log_size_prev)   9 2.841006 0.9469409  0.9725
## s(spei_history,L) 72 9.446201        NA      NA

Indicates default number of knots for log_spei_prev is adequate.

check_res_edf(s_1ha)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)     0

Knots are too low. Try upping knots for lag dimension

s_1ha2 <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(5, 30),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_1ha,
  method = "fREML",
  select = TRUE
)
edf(s_1ha)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   2.84
## 2 s(spei_history,L)  9.45
edf(s_1ha2)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   2.84
## 2 s(spei_history,L)  9.48
draw(s_1ha, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

draw(s_1ha2, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

No change, back to original k for lag and increase k for SPEI

s_1ha3 <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(10, 18),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_1ha,
  method = "fREML",
  select = TRUE
)
edf(s_1ha)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   2.84
## 2 s(spei_history,L)  9.45
edf(s_1ha3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   2.84
## 2 s(spei_history,L) 10.9
draw(s_1ha, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

draw(s_1ha3, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

Slight change, check if adequate

check_res_edf(s_1ha3)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)     0

Can I go lower?

s_1ha4 <- bam(
  surv ~ 
    flwr_prev +
    s(log_size_prev, bs = "cr") +
    s(spei_history, L,
      bs = "cb",
      k = c(10, 10),
      xt = list(bs = "cr")),
  family = binomial,
  data = model_data_1ha,
  method = "fREML",
  select = TRUE
)
edf(s_1ha3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   2.84
## 2 s(spei_history,L) 10.9
edf(s_1ha4)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   2.84
## 2 s(spei_history,L) 10.8
draw(s_1ha3, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

draw(s_1ha4, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

check_res_edf(s_1ha3)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)     0

10 knots all around

rm(s_1ha, s_1ha2, s_1ha3, s_1ha4)

Size

# use only living plants
model_data_1ha_2 <- model_data_1ha %>% 
  dplyr::filter(surv == 1, !is.na(log_size))
model_data_cf_2 <- model_data_cf %>% 
  dplyr::filter(surv == 1, !is.na(log_size))

Continuous forest

g_cf <- bam(log_size ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") + 
              s(spei_history, L,
                bs = "cb",
                k = c(5, 18), 
                xt = list(bs = "cr")),
            family = scat(link = "identity"), #like leptokurtic gaussian.
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
set.seed(888)
k.check(g_cf)
##                   k'       edf   k-index p-value
## s(log_size_prev)   9  5.702907 0.9765955    0.04
## s(spei_history,L) 72 15.694164        NA      NA

log_size_prev needs more knots.

g_cf2 <- bam(log_size ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr", k = 25) + 
              s(spei_history, L,
                bs = "cb",
                k = c(5, 18), 
                xt = list(bs = "cr")),
            family = scat(link = "identity"), #like leptokurtic gaussian.
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
set.seed(888)
k.check(g_cf2)
##                   k'       edf   k-index p-value
## s(log_size_prev)  24  9.370061 0.9805118  0.0825
## s(spei_history,L) 72 15.750214        NA      NA

Kept increasing until edf stabilized and p > 0.05. 25 knots is adequate.

How about the crossbasis?

check_res_edf(g_cf2)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)     0

Try reducing lag dimension knots to save on computation.

g_cf3 <- bam(log_size ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr", k = 25) + 
              s(spei_history, L,
                bs = "cb",
                k = c(5, 15), 
                xt = list(bs = "cr")),
            family = scat(link = "identity"), #like leptokurtic gaussian.
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
edf(g_cf2)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   9.37
## 2 s(spei_history,L) 15.8
edf(g_cf3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   9.37
## 2 s(spei_history,L) 15.7
draw(g_cf2, select = "s(spei_history,L)")
## Warning: Removed 968 rows containing non-finite values (stat_contour).

draw(g_cf3, select = "s(spei_history,L)")
## Warning: Removed 968 rows containing non-finite values (stat_contour).

Not substantially different

Final knots

  • log_size_prev: 25
  • spei_history, L: 5, 15

Fragments

g_1ha <- bam(log_size ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") + 
              s(spei_history, L,
                bs = "cb",
                k = c(5, 15), 
                xt = list(bs = "cr")),
            family = scat(link = "identity"), #like leptokurtic gaussian.
            data = model_data_1ha_2,
            method = "fREML",
            select = TRUE)
set.seed(888)
k.check(g_1ha)
##                   k'       edf  k-index p-value
## s(log_size_prev)   9  4.126322 1.013243    0.87
## s(spei_history,L) 60 19.230773       NA      NA

Default 10 knots for log_size_prev is adequate

check_res_edf(g_1ha)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)     0

Also adequate

draw(g_1ha,  select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

Final knots

  • log_size_prev: 10
  • spei_history, L: 5, 15
rm(g_cf, g_cf2, g_cf3, g_1ha)

Flowering

Continuous forest

f_cf <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(5, 18), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
set.seed(888)
k.check(f_cf)
##                   k'       edf   k-index p-value
## s(log_size_prev)   9  5.826355 0.9637795  0.7625
## s(spei_history,L) 72 11.237704        NA      NA

Knots for log_size_prev are adequate.

check_res_edf(f_cf)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)   1.5

Possibly too low. Try upping number of knots in the lag dimension.

f_cf2 <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(5, 30), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
edf(f_cf)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   5.83
## 2 s(spei_history,L) 11.2
edf(f_cf2)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   5.83
## 2 s(spei_history,L) 11.1

Not substantially different. Instead, try increasing knots in SPEI dimension

f_cf3 <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(15, 18), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
edf(f_cf)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   5.83
## 2 s(spei_history,L) 11.2
edf(f_cf3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   5.83
## 2 s(spei_history,L) 11.9
draw(f_cf, select = "s(spei_history,L)")
## Warning: Removed 968 rows containing non-finite values (stat_contour).

draw(f_cf3, select = "s(spei_history,L)")
## Warning: Removed 968 rows containing non-finite values (stat_contour).

Can I decrease knots in lag dimension?

f_cf4 <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(15, 15), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_cf_2,
            method = "fREML",
            select = TRUE)
edf(f_cf3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   5.83
## 2 s(spei_history,L) 11.9
edf(f_cf4)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   5.82
## 2 s(spei_history,L) 12.0

About the same

check_res_edf(f_cf4)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)   1.3

Final knots:

  • log_size_prev: 10
  • spei_history,L: 15, 15

Fragments

f_1ha <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(5, 18), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_1ha_2,
            method = "fREML",
            select = TRUE)
set.seed(888)
k.check(f_1ha)
##                   k'       edf   k-index p-value
## s(log_size_prev)   9  3.413398 0.9811181  0.8125
## s(spei_history,L) 72 12.826900        NA      NA

Knots for log_size_prev are adequate.

check_res_edf(f_1ha)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)  16.4

Too few knots for crossbasis. Try increasing knots in lag dimension.

f_1ha2 <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(5, 25), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_1ha_2,
            method = "fREML",
            select = TRUE)
edf(f_1ha)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.41
## 2 s(spei_history,L) 12.8
edf(f_1ha2)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.41
## 2 s(spei_history,L) 12.9

No substantial change. Set back to 18 and increase knots for SPEI dimension

f_1ha3 <- bam(flwr ~ 
              flwr_prev +
              s(log_size_prev, bs = "cr") +
              s(spei_history, L,
                bs = "cb",
                k = c(10, 18), 
                xt = list(bs = "cr")),
            family = binomial,
            data = model_data_1ha_2,
            method = "fREML",
            select = TRUE)
edf(f_1ha)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.41
## 2 s(spei_history,L) 12.8
edf(f_1ha3)
## # A tibble: 2 x 2
##   smooth              edf
##   <chr>             <dbl>
## 1 s(log_size_prev)   3.37
## 2 s(spei_history,L) 14.3
draw(f_1ha, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

draw(f_1ha3, select = "s(spei_history,L)")
## Warning: Removed 910 rows containing non-finite values (stat_contour).

(what I feared)

check_res_edf(f_1ha3)
## # A tibble: 1 x 2
##   smooth               edf
##   <chr>              <dbl>
## 1 te(spei_history,L)     0

adequate knots.

Final knots:

  • log_size_prev: 10
  • spei_history,L: 10,18
rm(f_cf, f_cf2, f_cf3, f_cf4, f_1ha, f_1ha2, f_1ha3)

Reproducibility

Reproducibility receipt 
## datetime
Sys.time()
## [1] "2021-08-17 17:08:26 EDT"
## repository
if(requireNamespace('git2r', quietly = TRUE)) {
  git2r::repository()
} else {
  c(
    system2("git", args = c("log", "--name-status", "-1"), stdout = TRUE),
    system2("git", args = c("remote", "-v"), stdout = TRUE)
  )
}
## Local:    revisions /Users/scottericr/Documents/HeliconiaDemography
## Remote:   revisions @ origin (https://github.com/BrunaLab/HeliconiaDemography.git)
## Head:     [d40aa88] 2021-08-17: function to plot "slices" through 2d smooths
## session info
sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS  10.16
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats4    parallel  stats     graphics  grDevices datasets  utils    
## [8] methods   base     
## 
## other attached packages:
##  [1] qqplotr_0.0.5     Hmisc_4.5-0       Formula_1.2-4     survival_3.2-11  
##  [5] lattice_0.20-44   readxl_1.3.1      colorspace_2.0-1  rmarkdown_2.7    
##  [9] statmod_1.4.36    latex2exp_0.5.0   gratia_0.6.0.9112 broom_0.7.6      
## [13] patchwork_1.1.1   glue_1.4.2        bbmle_1.0.23.1    dlnm_2.4.5       
## [17] mgcv_1.8-36       nlme_3.1-152      lubridate_1.7.10  janitor_2.1.0    
## [21] tsModel_0.6       SPEI_1.7          lmomco_2.3.6      tsibble_1.0.1    
## [25] forcats_0.5.1     stringr_1.4.0     dplyr_1.0.5       purrr_0.3.4      
## [29] readr_1.4.0       tidyr_1.1.3       tibble_3.1.1      ggplot2_3.3.5    
## [33] tidyverse_1.3.1   here_1.0.1        tarchetypes_0.2.0 targets_0.4.2    
## [37] dotenv_1.0.3      conflicted_1.0.4 
## 
## loaded via a namespace (and not attached):
##  [1] ellipsis_0.3.2      rprojroot_2.0.2     htmlTable_2.1.0    
##  [4] snakecase_0.11.0    base64enc_0.1-3     fs_1.5.0           
##  [7] rstudioapi_0.13     farver_2.1.0        fansi_0.4.2        
## [10] mvtnorm_1.1-1       xml2_1.3.2          codetools_0.2-18   
## [13] splines_4.0.2       Lmoments_1.3-1      robustbase_0.93-7  
## [16] cachem_1.0.4        knitr_1.33          jsonlite_1.7.2     
## [19] anytime_0.3.9       cluster_2.1.2       dbplyr_2.1.1       
## [22] png_0.1-7           compiler_4.0.2      httr_1.4.2         
## [25] backports_1.2.1     assertthat_0.2.1    Matrix_1.3-3       
## [28] fastmap_1.1.0       cli_2.5.0           htmltools_0.5.1.1  
## [31] tools_4.0.2         igraph_1.2.6        gtable_0.3.0       
## [34] Rcpp_1.0.6          jquerylib_0.1.4     cellranger_1.1.0   
## [37] vctrs_0.3.8         xfun_0.22           ps_1.6.0           
## [40] rvest_1.0.0         lifecycle_1.0.0     renv_0.13.2        
## [43] goftest_1.2-2       DEoptimR_1.0-8      MASS_7.3-54        
## [46] scales_1.1.1        hms_1.1.0           RColorBrewer_1.1-2 
## [49] yaml_2.2.1          mvnfast_0.2.5.1     gridExtra_2.3      
## [52] memoise_2.0.0       sass_0.3.1          rpart_4.1-15       
## [55] bdsmatrix_1.3-4     latticeExtra_0.6-29 stringi_1.6.2      
## [58] highr_0.9           checkmate_2.0.0     rlang_0.4.11       
## [61] pkgconfig_2.0.3     evaluate_0.14       labeling_0.4.2     
## [64] htmlwidgets_1.5.3   processx_3.5.2      tidyselect_1.1.1   
## [67] magrittr_2.0.1      R6_2.5.0            generics_0.1.0     
## [70] DBI_1.1.1           pillar_1.6.0        haven_2.4.1        
## [73] foreign_0.8-81      withr_2.4.2         nnet_7.3-16        
## [76] modelr_0.1.8        crayon_1.4.1        utf8_1.2.1         
## [79] jpeg_0.1-8.1        isoband_0.2.4       grid_4.0.2         
## [82] data.table_1.14.0   git2r_0.28.0        callr_3.7.0        
## [85] reprex_2.0.0        digest_0.6.27       numDeriv_2016.8-1.1
## [88] munsell_0.5.0       bslib_0.2.4